4.1.6. Comt: comptonisation model

This is the model for Comptonisation of soft photons in a hot plasma, developed by Titarchuk (1994). See the XSPEC manual for more details. The code was substantially improved and brought to the SPEX standards.

4.1.6.1. Some modifications

Titarchuk (1994) gives an analytical approximation for \beta(\tau) to the exact calculations as provided by Sunyaev & Titarchuk (1985) for 0.1<\tau<10. Their approximation has the proper asymptotic behaviour for small and large values of \tau, but unfortunately has deviations up to 7 % over the 0.1-10 range for \tau. The approximation by Titarchuk is given in their equations (27) and (28) by

\begin{aligned}
\mbox{disk:} \beta &=&
\frac{\pi^2}{12(\tau+2/3)^2} (1-{\mathrm e}^{\displaystyle{-1.35\tau}})
+0.45{\mathrm e}^{\displaystyle{-3.7\tau}}\ln \frac{10}{3\tau}, \\
\mbox{sphere:} \beta &=&
\frac{\pi^2}{ 3(\tau+2/3)^2} (1-{\mathrm e}^{\displaystyle{-0.7\tau}})
+{\mathrm e}^{\displaystyle{-1.4\tau}}\ln \frac{4}{3\tau}.\end{aligned}

We found an approximation that is better than 1 % using a slight modification of the above equations. We use these new approximations in our calculations:

\begin{aligned}
\mbox{disk:} \beta &=&
\frac{0.8351}{(\tau+0.867)^2}(1-{\mathrm e}^{\displaystyle{-3.876\tau}})
+0.572{\mathrm e}^{\displaystyle{-3.75\tau}}\ln \frac{1.06}{\tau}, \\
\mbox{sphere:} \beta &=&
\frac{3.504}{(\tau+1.2)^2}(1-{\mathrm e}^{\displaystyle{-2.09\tau}})
+1.013{\mathrm e}^{\displaystyle{-2.2\tau}}\ln \frac{1.6}{\tau}.\end{aligned}

Warning

For the physically valid range of parameters, consult the XSPEC manual; see also Hua & Titarchuk (1995), their Fig. 7.

The parameters of the model are:

norm : Normalisation A of the power law, in units of 10^{44} \mathrm{ph} \mathrm{s}^{-1} \mathrm{keV}^{-1}. Due to the way the model is calculated, this is not the flux at 1 keV! Default value: 1.
t0 : The temperature T of the seed photons in keV. Default value: 1 keV.
t1 : The plasma temperature T in keV. Default value: 100 keV.
tau : The optical depth. Default value: 3.
type : The type of geometry. Allowed values are 0 for a disk (default) or 1 for a sphere. This is a frozen parameter (cannot be fitted).

Recommended citation: Titarchuk (1994)